What is the largest double word square or rectangle (containing a
different English word in each row and column) that can be formed from
the 98 non-blank Scrabble tiles? The purpose of this article is explore,
and give the answers to, questions like this in which the common
word-play form of the word rectangle is constrained by the number of
tiles of each letter available in the game of Scrabble, which is:

In many logological investigations the question "What should be
considered a word?" arises. In this case the answer seems obvious
and natural, which is to permit just those words that are legal in the
game of Scrabble. For this reason I chose to use the current (6th)
version of the Official Tournament and Club Word List (often called TWL)
which is based on the Official Scrabble Players Dictionary (OPSD).
Readers wishing to duplicate these results or make further
investigations can, with a bit of searching, find this word list online
(it is usually called TWL06.TXT).

I also decided to require that all the word rectangles be
"dense", in which all of the words in the rows and columns are
distinct. It might be guessed that this is not much of a restriction,
but there are a surprising number of even large rectangles with repeated
words, such as this 7x6 (note the top and bottom row):

MADDEST ARIETTA REACHES ROCKERS ALIENEE MADDEST

The same rule applies when the rectangle is a square, which in
addition to disallowing squares like the rectangle above also prohibits
"single" word squares, in which every row word is the same as
the corresponding column word.

The search space that much be examined to find all possible Scrabble
word rectangles of all possible sizes is, it turns out, small enough to
search exhaustively, but only if an efficient search algorithm is used.
The primary key to a fast search is to store the word list not as a
simple list of words but rather in a data structure known as a trie
(from the phrase "fast retrieval"). The time-consuming
operation in an exhaustive search is illustrated by the figure below, in
which some letters have been filled in already and the question being
asked is: which letters of the alphabet are permissible in the place
marked "x"?

BRIG AUTO Sx.. ....

With words stored in a trie, the legal possibilities for
"x" (whose choice produces prefixes RUx and Sx that both occur
in the dictionary) can be determined using just a few computer
operations, hundreds of times faster than would be required with the
dictionary stored as a linear list of words.

Using about 20 hours of time on a home computer I was able to find
all word squares and rectangles of all possible sizes and store them in
files. The table below shows the exact number of dense Scrabble word
rectangles of all possible sizes:

Summing all the numbers in above table leads gives a grand total of
342,286,026 rectangles. Note that in counting the squares (not the
rectangles) one needs to be careful to count only one of each transpose
pair.

The table below shows the unique 7x7, the three 8x6's (two of
which are very similar), one example each of 9x5, 10x4, 12x3, and the
unique 13x2; these have the largest possible width for each height.

The 7x7 answers the question posed in the introduction, as it is the
(unique!) double square or rectangle having the largest area (49). In
fact, it is not possible to make a square or rectangle larger than this
out of the Scrabble tiles even if the two blank tiles are permitted.

Once we have all the double rectangles enumerated and stored on the
computer, it is a simple matter to scan through them and find those with
various special properties. For example, it is natural to ask which ones
have letters that sum to the highest Scrabble score. Here are the top
eight:

The smallest size having a unique highest-scoring grid is 6x2; of
the 72,093 6x2's, only one achieves the maximum score of 37. The
most populous size with a unique highest-scoring grid is 4x4; of the
14,592,390 4x4's, only one achieves the maximum score of 53. These
two special grids are shown below.

SHAZAM HEXADE (37)
CHOW HOYA IBEX COZY (53)

The largest grids containing all one-point Scrabble tiles are 6x6,
there are three such:

The largest grid with a perfect checkerboard of vowels and
consonants (with Y classified as a vowel) is the 7x6 shown on the left
below. Exactly one of the 31 million 5x5's, the one shown on the
right below, has a different remarkable feature: each letter on a black
square of the checkerboard has a point value of one, and each letter on
a red square has a point value greater than one. This is the largest
square or rectangle with this property.

It is not too surprising to find that there are no pangram grids of
any size. There are, however, two grids that contain 21 different
letters of the alphabet (the record), with just P, V, X, Y and Z
missing; they are

JACKSHAFT AQUILEGIA MURDERING BASSWOODS

and the same grid with the third line changed to MUTTERING.

Define the "diversity" of a word rectangle as (number of
different letters) / (total number of letters). The unique grid with the
smallest diversity (0.2) is the 5x4 shown below, which only uses the 4
letters A, E, R, S:

ERASE RARES AREAE SEARS

There are no grids of any size containing all four of the rare
letters J, Q, X, Z, but there are many containing three of the four; the
largest is this 7x5 grid:

QUIVERS ALLOXAN ITEMIZE DRAINED SALTERS

The unique grid having the largest number of palindromes (eight, all
in the columns) is this 9x3:

ANTEDATES GUAYABERA ANTEDATED

The longest palindrome that occurs in any square or rectangle is 7
letters long. Amazingly, there is a grid containing two such
palindromes:

HALALAH ALAMEDA REVIVER PRECEPT STREETS

This 9x5 is the unique grid with the largest number of distinct
words contained in the rows and columns (reading only right or down is
allowed, not left or up):

What is the largest grid that contains at least one full-length word
on some line going diagonally down and to the right? The answer is 9x5,
and there are two such grids, shown below with the diagonal words (which
just happen to be in the same place in both grids) underlined. The
largest square of this kind is 6x6 and there are 128 of these, one of
which is displayed below.